| L(s) = 1 | − 5.37·3-s + 19.8·9-s − 11·11-s − 8.35·23-s − 58.3·27-s − 24.5·31-s + 59.0·33-s − 72.8·37-s − 50·47-s + 49·49-s + 70·53-s − 96.5·59-s + 129.·67-s + 44.8·69-s + 23.4·71-s + 134.·81-s + 177.·89-s + 132.·93-s + 193.·97-s − 218.·99-s + 190·103-s + 391.·111-s + 47.1·113-s + ⋯ |
| L(s) = 1 | − 1.79·3-s + 2.20·9-s − 11-s − 0.363·23-s − 2.16·27-s − 0.793·31-s + 1.79·33-s − 1.96·37-s − 1.06·47-s + 0.999·49-s + 1.32·53-s − 1.63·59-s + 1.93·67-s + 0.650·69-s + 0.329·71-s + 1.66·81-s + 1.99·89-s + 1.42·93-s + 1.99·97-s − 2.20·99-s + 1.84·103-s + 3.52·111-s + 0.417·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6233319650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6233319650\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 5.37T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 + 8.35T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 24.5T + 961T^{2} \) |
| 37 | \( 1 + 72.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 + 50T + 2.20e3T^{2} \) |
| 53 | \( 1 - 70T + 2.80e3T^{2} \) |
| 59 | \( 1 + 96.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 129.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 23.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 177.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 193.T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.09843183863451340322153373717, −8.930355628359587040206443461966, −7.75527311470485631476912447363, −7.00258087939513476838501169801, −6.15595713211672844628714387583, −5.35938410835175257032648058757, −4.82327039995969659584999305033, −3.61832489622515076994631271917, −1.95140151249360916407087244599, −0.51560464564822244241546287375,
0.51560464564822244241546287375, 1.95140151249360916407087244599, 3.61832489622515076994631271917, 4.82327039995969659584999305033, 5.35938410835175257032648058757, 6.15595713211672844628714387583, 7.00258087939513476838501169801, 7.75527311470485631476912447363, 8.930355628359587040206443461966, 10.09843183863451340322153373717
